Let's add markers on the diagram to indicate that ∠ 1≅∠ 3, which is given in the question.
Checking F and G
The conclusions in option F and G are very similar, so let's investigate what is implied if both of these are true.
F:& m∠ 1-m∠ 2+m∠ 3=90
G:& m∠ 1+m∠ 2+m∠ 3=180
We can see that m∠ 1+m∠ 3 appear in both of these equation. If we subtract the first equation from the second, both m∠ 1 and m∠ 3 are eliminated.
2m∠ 2=90 ⟺ m∠ 2=45
Since the measure of ∠ 2 is not fixed, this means that the conclusions in F and G do not have to both be true. Let's check the conclusion in G.
Checking G
G:& m∠ 1+m∠ 2+m∠ 3=180
∠ 1, ∠ 2, and ∠ 3 are adjacent angles, and together they form a . The Angle Addition Postulate guarantees that the sum of their angles is the measure of the straight angle, which is 180. Thus, the conclusion in G is always true.
Conclusion
By now we know that:
- the conclusions in F and in G together are not always true, and
- the conclusion in G is always true.
These imply that the conclusion in F does not have to be true. The correct choice is F.
Notice that in the solution above we did not use the assumption about the congruence of ∠ 1 and ∠ 3. This assumption is needed for showing that the conclusions in H and J are always true. These can be proved using the definition of congruence.
